Time-Varying Dispersion Integer-Valued GARCH Models

We propose a general class of INteger-valued Generalized AutoRegressive Conditionally Heteroskedastic (INGARCH) processes by allowing time-varying mean and dispersion parameters, which we call time-varying dispersion INGARCH (tv-DINGARCH) models. More specifically, we consider mixed Poisson INGARCH models and allow for a dynamic modeling of the dispersion parameter (as well as the mean), similarly to the spirit of the ordinary GARCH models. We derive conditions to obtain first and second order stationarity, and ergodicity as well. Estimation of the parameters is addressed and their associated asymptotic properties established as well. A restricted bootstrap procedure is proposed for testing constant dispersion against time-varying dispersion. Monte Carlo simulation studies are presented for checking point estimation, standard errors, and the performance of the restricted bootstrap approach. The inclusion of covariates is also addressed and applied to the daily number of deaths due to COVID-19 in Ireland. Insightful results were obtained in the data analysis, including a superior performance of the tv-DINGARCH processes over the ordinary INGARCH models.